Let's break down the solution step by step as per your instructions: --- ## Step 1: **Conceptual Introduction** **Conceptual Introduction:** This problem involves analyzing fluid flow between two tanks connected by a pipeline system with multiple components (valves, elbows, etc.) and a pump. The height difference between tanks and pipeline losses must be considered. Using principles from fluid mechanics, particularly the energy equation (Bernoulli’s equation), we can determine the flow rate and suitable pump selection for the system. **Explanation Block:** The system comprises open tanks linked by a 25 m long, 20 cm diameter pipe. The pump must overcome both elevation (height) difference and all losses (major and minor) in the pipeline. By applying the extended Bernoulli equation, which accounts for pump head, frictional losses, and minor losses, we can relate the physical setup to the flow rate and pump requirements. --- ## Step 2: **Formulae Used** **Formulae:** 1. **Extended Bernoulli’s Equation (between tank surfaces):** \[ h_{\text{pump}} = \Delta z + h_f + h_L \] - \( h_{\text{pump}} \): Pump head (m) - \( \Delta z \): Elevation difference (m) - \( h_f \): Major losses (friction losses in pipe) (m) - \( h_L \): Minor losses (due to valves, elbows, etc.) (m) 2. **Darcy-Weisbach Equation for Major Losses:** \[ h_f = f \frac{L}{D} \frac{V^2}{2g} \] 3. **Minor Losses:** \[ h_L = \sum K_L \frac{V^2}{2g} \] - \( K_L \): Loss coefficients for each fitting 4. **Flow Rate:** \[ Q = AV = \frac{\pi D^2}{4} V \] **Explanation Block:** The extended Bernoulli equation sums all energy changes between tanks, including pump work, height difference, and all losses. Major losses are calculated using the Darcy-Weisbach equation, which accounts for pipe length, diameter, flow velocity, and friction factor. Minor losses are summed from all fittings using their respective loss coefficients. The flow rate is related to velocity and pipe diameter. --- ## Step 3: **Step-by-Step Calculation for Part (a) – Flow Rate** ### **Step 1:** List known values. - \( D = 0.2 \) m - \( L = 25 \) m - \( \Delta z = 11 \) m - \( f = 0.02 \) - \( g = 9.81 \) m/s² **Minor losses:** - Entrance loss (\( K_L = 0.6 \)) - Pump suction (\( K_L = 1.4 \)) - Exit loss (\( K_L = 0.9 \)) - 4 × 90° elbow (\( 4 \times 0.3 = 1.2 \)) - Check valve (\( 2 \)) - Globe valve (\( 10 \)) - Tee (\( 0.2 \)) Sum all \( K_L \): \[ K_{L,\text{total}} = 0.6 + 1.4 + 0.9 + 1.2 + 2 + 10 + 0.2 = 16.3 \] ### **Step 2:** Express all losses in terms of velocity (\( V \)). #### Major Loss: \[ h_f = f \frac{L}{D} \frac{V^2}{2g} = 0.02 \times \frac{25}{0.2} \frac{V^2}{2 \times 9.81} = 2.5 \frac{V^2}{2 \times 9.81} \] \[ h_f = 2.5 \frac{V^2}{19.62} = 0.1275 V^2 \] #### Minor Loss: \[ h_L = K_{L,\text{total}} \frac{V^2}{2g} = 16.3 \frac{V^2}{19.62} = 0.831 V^2 \] #### Total Losses: \[ h_{\text{total losses}} = h_f + h_L = 0.1275 V^2 + 0.831 V^2 = 0.9585 V^2 \] ### **Step 3:** Write pump head equation. \[ h_{\text{pump}} = \Delta z + h_{\text{total losses}} = 11 + 0.9585 V^2 \] ### **Step 4:** Relate velocity to flow rate. \[ Q = AV = \frac{\pi}{4} D^2 V = 0.0314 V \] \[ V = \frac{Q}{0.0314} \] ### **Step 5:** Substitute for \( V \) in pump head equation. \[ h_{\text{pump}} = 11 + 0.9585 \left(\frac{Q}{0.0314}\right)^2 \] \[ h_{\text{pump}} = 11 + 0.9585 \times \frac{Q^2}{0.00098596} \] \[ h_{\text{pump}} = 11 + 972.5 Q^2 \] ### **Step 6:** Use pump characteristic curve (from FIGURE A2, not provided), or state the relationship to solve for \( Q \) at the intersection of the system and pump curve. **Explanation Block:** All losses (major and minor) are expressed in terms of velocity, then related to the flow rate using the pipe cross-section. The total head the pump must provide is the sum of elevation and all losses. Using the pump curve (not shown here), the intersection point with the system curve gives the actual flow rate. --- ## Step 4: **Step-by-Step Calculation for Part (b) – Pump Selection** ### **Step 1:** Consider pump location. Pump inlet is 3 m above Tank 1 water surface (this affects NPSH available, but not the head required between tanks). ### **Step 2:** Use the system curve from above: \[ h_{\text{pump}} = 11 + 972.5 Q^2 \] ### **Step 3:** Refer to FIGURE A2 (not provided). Select pump whose curve matches the system curve at a reasonable operating point. Typically, choose the pump whose curve intersects the system curve at or near its best efficiency point. ### **Step 4:** Justify selection. Choose the pump that delivers the required head at the calculated flow rate, and ensure it is capable of operating with the inlet 3 m above the tank level (check NPSH if data available). **Explanation Block:** Pump selection is based on matching the required head (system curve) with the pump’s performance curve. The operating point where both curves intersect determines the actual flow and head. The pump must also be able to operate safely with its inlet elevated above the water surface, avoiding cavitation. --- ## **Final Answer (Summary):** - **a.** The flow rate is found by solving \( h_{\text{pump}} = 11 + 972.5 Q^2 \) using the pump characteristic curve. - **b.** Select the pump whose characteristic curve intersects the system curve at the calculated operating point, and ensure it is suitable for installation 3 m above the tank water surface. **Final Answer:** **The flow rate is obtained where the pump curve meets the system curve \( h_{\text{pump}} = 11 + 972.5 Q^2 \). Choose a pump from FIGURE A2 that delivers this head at the corresponding flow rate and is suitable for installation 3 m above the tank.**